The Behavior of Tree-Width and Path-Width under Graph Operations and Graph Transformations

TL;DR

This paper investigates how various graph transformations affect tree-width and path-width, providing bounds and insights crucial for understanding the complexity of graph problems under these operations.

Contribution

It offers a comprehensive analysis of the behavior of tree-width and path-width under numerous graph transformations, establishing bounds and impossibility results.

Findings

Upper and lower bounds for tree-width and path-width after transformations

Identification of transformations that preserve or significantly alter these parameters

Insights into the complexity implications for graph algorithms

Abstract

Tree-width and path-width are well-known graph parameters. Many NP-hard graph problems allow polynomial-time solutions, when restricted to graphs of bounded tree-width or bounded path-width. In this work, we study the behavior of tree-width and path-width under various unary and binary graph transformations. Doing so, for considered transformations we provide upper and lower bounds for the tree-width and path-width of the resulting graph in terms of the tree-width and path-width of the initial graphs or argue why such bounds are impossible to specify. Among the studied, unary transformations are vertex addition, vertex deletion, edge addition, edge deletion, subgraphs, vertex identification, edge contraction, edge subdivision, minors, powers of graphs, line graphs, edge complements, local complements, Seidel switching, and Seidel complementation. Among the studied, binary transformations we consider the disjoint union, join, union, substitution, graph product, 1-sum, and corona of two graphs.